7. THE QUASAR LUMINOSITY FUNCTION: CURRENT STATUS AND FUTURE PROSPECTS

The rationale for much survey work has been the determination of how
the quasar population evolves as a function of look-back time.
Following the pioneering work of Schmidt in the late 1960s, and the
analysis of larger samples in the 1980s by Marshall, Boyle, Crampton,
Usher and others, the qualitative picture is well established. For
z 2,
samples defined at radio, optical and X-ray
wavelengths show a rapid increase in space density as a function of
look-back time (Boyle
1993, Maccacaro
et al. 1991,
Dunlop and Peacock
1990). Over the redshift range 0 z 2 the rate of
evolution inferred from optical samples corresponds to an increase in
integral space density of a factor 200 (for quasars with
MB = -26), or, equivalently, to the characteristic
luminosity of the
quasar population increasing by a factor of ~ 40. A feature of all
the analyses is the simplicity of the model consistent with the data at
z 2: a two
power-law luminosity function, of invariant shape,
whose characteristic luminosity evolves as a function of redshift
according to L* (1 +
z)k. The shape of the luminosity
function shows quantitative differences between the radio, optical and
X-ray, and the rate of evolution is somewhat different, with values of
k = 3.0, 3.45, 2.75 (q0 = 0.5) for radio,
optical and X-ray samples
respectively (Boyle
1993). However, the simplicity of the model and the
similarity between samples defined at different wavelengths is
striking. At redshifts z >2 few data are available in the radio and
X-ray regimes, and only recently have suitable samples become
available at optical wavelengths, however, it is clear that the rate of
evolution at redshifts z > 2 slows dramatically. At
z > 3, there is no consensus as to whether the space
density continues to increase or whether a decline has begun. In the
remainder of this section, we discuss areas where recent advances have produced
new results, or where current work is expected to provide significantly
improved constraints.

For a decade the analysis of the Palomar-Green sample
(Schmidt and Green
1983) has stood as the reference for studies of high-luminosity
quasars at the brightest apparent magnitudes, mB 16. The
existence of redshift-dependent selection effects and substantial
uncertainties in the Palomar-Green magnitudes have been known for some
time (Wampler and
Ponz 1985), but it is only recently that additional
data at bright magnitudes have become available. The Edinburgh
multicolor quasar survey,
(Goldschmidt et
al. 1992) shows that the
surface density of for mB 16 is a factor 2-3
higher than in
the Palomar-Green sample. This conclusion is supported by the analysis
of the LBQS (Hewett,
Foltz and Chaffee 1993), which covers the
magnitude range 16.5 mBJ 18.85, and also indicates an
excess of luminous quasars at low redshifts. In a second paper
Miller et al.
(1993) take the more extreme view that the space density of the
most luminous quasars, MB ~ -29, has not changed over
the entire
redshift range 0.2 < z < 4. While this interpretation is still
controversial, a quantitative departure from the standard model
evident from the Edinburgh survey and the LBQS is the systematic
steepening of the bright end of the luminosity function with
increasing redshift. These analyses do not include the
selection function
calculations so they should be considered preliminary, but the results
appear relatively secure based on the agreement in the
form of the source counts with previous work and the smooth redshift
distributions of the samples.
The publication of these two surveys has increased the
number of quasars with 16.5 < mB < 18.5 in
well-defined samples by
more than an order of magnitude.

The motivation behind finding objects at high redshifts, derives from
the large look-back times, ,
at redshift z = 4; for
example, = 0.93 for
q0 = 0.5, = 0. Notwithstanding the
recent advances made in the detection of radio galaxies at
high-redshifts, quasars remain the only population detected in
significant numbers at z > 4. The extreme nature of quasars
makes their relevance to our understanding of galaxy formation a matter
of debate, but Efstathiou and Rees (1988), and more recently
Haehnelt and Rees
(1993), have used the evolution of the quasar luminosity
function to place constraints on the formation of massive, ~
1011-1012M, bound systems at early
epochs. An important
by-product of locating apparently bright quasars at z > 4 is the
ability to study the behavior of intervening absorption systems at
very high redshifts. Investigations of the Lyman- forest,
Lyman-limit, and damped Lyman- systems are in progress, using
quasars located by Irwin, McMahon and Hazard (1991).

At the highest redshifts, locating quasars at optical wavelengths
becomes even more difficult. The surface densities relative to
contaminating objects (Section 3.1)
decrease by an additional one or two orders
of magnitude. The shift of the ultraviolet continuum through the
optical passbands means that shortward of 5500 Å the observed flux
is reduced dramatically by the presence of the increasingly prevalent
Lyman- forest lines and
the higher column density metal-line
absorption systems. Working in the R band ( ~ 6500 Å) or
at longer wavelengths is a prerequisite for success, but even at red
wavelengths, luminous quasars are apparently faint - an
MB = -26
quasar has an apparent magnitude mR ~ 20.5 at z = 4.

At the time of the first searches for quasars with redshifts z ~ 4,
it was not known that the strong evolution at redshifts
z ~ 2 did not extend to higher redshifts. Based on the assumption
that the evolution continued, large numbers of faint, high-redshift
quasars were predicted, and searches concentrated on small areas of
sky, probing to relatively faint magnitudes. Following
Osmer's (1982)
paper, the failure to identify high-redshift quasars from several
surveys confirmed that the rapid rate of evolution observed at
low-redshift must decrease above redshift z ~ 2. With this
knowledge investigations probing brighter magnitudes over much
larger areas of sky were initiated:
Schmidt, Schneider
and Gunn (1991)
employing their CCD-based grism technique to a larger area,
Warren et al.
(1991a) using broadband multicolor selection, and
Irwin, McMahon and
Hazard (1991) utilizing a highly-specific version of the
multicolor technique - essentially a high-redshift version of the
ultraviolet excess technique. The parameters defining the latter of
these, an area of 2500 deg2 to an apparent magnitude
mR ~ 19,
cf. <1 deg2 to mR ~ 20.5
(Schmidt, Schneider
and Gunn 1986a),
illustrate how the experimental design has evolved. The Irwin, McMahon
and Hazard survey has proved outstandingly successful in identifying
z > 4 quasars, with some 27 objects now confirmed. Coupled with
the derivation
of the first quantitative space density estimates by
Schmidt et al.
(1994) and Warren
et al. (1994) the investigation of the high-redshift
regime may be regarded as something of a success.

Warren et al.
(1994) conclude there is a substantial decline in the
space density between z ~ 3.3 and z 4. If the
extent of the decline in space density at high redshift is as large as
they claim, the prospects for identifying many quasars with z > 5 are
poor. However, the constraints on the amplitude of the decline are
still weak and taking the least extreme model consistent with the data
suggests that searches of ~ 100 deg2, to a limiting magnitude of
mI ~ 20, may detect a number of z > 5 quasars. Even if
no objects were found, the improved constraints on the rate of decline
would be very valuable. At redshifts much greater than z = 5 it is
tempting to consider searches using near-infrared arrays; at z = 6,
Lyman- appears at = 8500 Å and Mg II
2798 is nearly into the
K-band. If evidence were found
that observed
quasar SEDs become significantly redder due (say) to
extinction then the large I - K colors would make such an approach
worth considering. However, such evidence remains elusive and, given the
limited size of infrared arrays, the bright sky-background and the
modest I - K colors of quasars with typical SEDs, the near-infrared
is unlikely to provide the means to undertake major surveys over large
areas in the near future.

The understanding of the evolution of the quasar X-ray
luminosity function has been revolutionized by the completion of two
surveys: the Extended Medium Sensitivity Survey (EMSS), and very deep
ROSAT observations of the Boyle et al. (1990) ultraviolet
excess survey
fields. Maccacaro and collaborators have defined a sample of 427
X-ray-selected quasars and AGN observed by the Einstein satellite.
These objects were selected following the spectroscopic identification
of the flux-limited catalog of the EMSS
(Gioia et al.
1990). The
Einstein satellite was sensitive to the energy range 0.3-3.5
keV and extends to a flux level of 10-13 erg s-1
cm-2. This corresponds to the flux expected from a rather
luminous X-ray quasar, Lx ~ 1045 erg
s-1 at
redshifts z > 1, but sensitivity at this flux level also enables the
luminosity function to be probed as faint as Lx ~
1042 erg s-1 locally. The sample is the X-ray
equivalent
of the optical Palomar-Green survey, in that it has provided the
reference for defining the properties of the X-ray luminosity function
at low-redshifts and high luminosities. The EMSS is well-suited to
the analysis of the quasar population, since the flux-limits are
well-defined, and the spectroscopic identification is (almost)
complete for objects within these limits. The analysis is described in
Maccacaro et
al. (1991) and Della Ceca and Maccacaro (1991).

For z 1,
constraints on the evolution of the
luminosity function require a sample extending to faint flux levels,
such as that constructed by Shanks, Boyle and collaborators
(Boyle et al.
1993), who exploited the excellent match between
the ROSAT
satellite's field of view and the constituent fields making up the
Boyle et al.
(1990) ultraviolet excess survey. Deep ROSAT
exposures, with flux limits of
~ 6 x 10-15 erg s-1 cm-2 in the
0.5-2.0 keV energy range, in several of their optical fields
have produced an X-ray selected catalog of 42 confirmed quasars, the
majority of which have z > 1. Twelve of the sources in the
ROSAT exposures remain unidentified. Combining the EMSS and ROSAT
samples Boyle et
al. (1993) find that a two-power-law form for the
luminosity function, that is invariant in shape and whose
characteristic luminosity evolves as Lx* (1 + z)2.8
± 0.1 (q0 = 0.0) fits the data well. A
significant
slow-down in the rate of evolution at z ~ 2 is also required. The
similarity with the form of the evolution in the optical is striking.

There are differences between the Boyle et al. (1993)
analysis of the combined EMSS and ROSAT data and the Maccacaro
et al.
and Della Ceca and Maccacaro analyses of the EMSS sample alone. Boyle
et al. address this problem explicitly, ascribing part of the difference
to the methods of analysis and to the improved constraints arising from
combining the samples. However, a key factor is how the two samples
are related. The different energy ranges to which the Einstein and
ROSAT detectors are sensitive, combined with the higher mean redshift
of the ROSAT sample, means both the shape of the mean quasar SED at
X-ray frequencies and the distribution of SEDs about the mean must be
known. Approximating the X-ray SED of the quasars using a power-law
in frequency, Boyle et al. test for the effect of a dispersion in the
power-law slopes, and also whether inclusion of the unidentified
objects could affect the results, and conclude that neither is
important. The tests assume no correlation between redshift, quasar
SED and the probabilities that a quasar remains unidentified. Thus,
while the tests as described are valid, they are equivalent to assuming
the array of probabilities P (M, z, SED) may be written
P (M)·P (z)·P (SED),
and further, that P (z), for example, is
constant. Given the lack of information on the quasar SEDs it
is sensible to assume that the selection function is separable, but
there is no evidence to support such a contention. Once again, the
critical factor limiting our knowledge is the lack of information about
the form and distribution of quasar SEDs and the possible variations of
the distribution as a function of M and z.

The difference in approach to the analysis of surveys propounded in
this review and that of Boyle et al. is encapsulated in the statement at
the end of their Section 4: ``Certainly,
samples of QSOs at X-ray
wavelengths are much more useful for providing an unequivocal
determination of the existence of a cutoff at z ~ 2, since they are
significantly less prone to the selection effects which bedevil optical
samples of QSOs at z > 2.'' In contrast, we believe that the lack of
information concerning quasar SEDs at both optical and X-ray wavelengths
is the limiting factor, and that once a determination of the selection
function is made for surveys at both wavelengths the conclusions
regarding the evolution of the luminosity function and the intrinsic
distribution of SEDs should be equally robust and entirely consistent.

Throughout this review we have stressed that a factor limiting our
knowledge of the quasar population and its evolution is the lack of
information concerning quasar SEDs.
Approximating quasar SEDs by a
power-law in frequency, ,
F ()
, the relation between
the apparent magnitude,
mB, and the absolute magnitude, MB, is:

MB = mB - 5 log [A (z) c /
H0] + 2.5 (1 + ) log(1 + z) - 25,

where A (z) c / H0 is the luminosity distance
(Schmidt and Green
1983).
The discussion in this section assumes apparent and absolute magnitudes
are referenced to the B band, but the principles apply equally to
other wavebands, or indeed, to apparent
magnitudes and absolute magnitudes referenced to different wavelengths,
e.g., absolute magnitudes MB derived from apparent
magnitudes in the
R band, mR. An error, (), in the mean spectral
index of the population translates into an error in the derived
absolute magnitude. The size of the latter increases with redshift,
resulting in a spurious ``evolution'' of the quasar population. Over
the redshift interval 0 < z < 1, an error of () = 0.5
in the mean spectral index, will result in the absolute magnitude
calculated for a quasar at z = 1 being in error by MB = 0.37. At redshift z = 2 this increases to
MB =
0.60.
If the characteristic luminosity of the luminosity function
evolves as L* (1 +
z)k, the error in the spectral index,
(), affects the evolution
parameter, k, such that,
k' = k - (), where
k' is the inferred value of the
evolution parameter. Alternatively, note that reaching 0.6 magnitudes
fainter into
the bright end of the quasar luminosity function, where N(L)L-3.5, at z = 2 produces an increase of a factor ~ 7
in number, and hence the same factor in space density.
Errors of several tenths in the effective mean power-law index of quasar
SEDs are quite possible, and Boyle et al. (1993) note that such errors
may in part be responsible for the differing rates of evolution
observed in the optical and X-ray regimes, and also between
samples defined at different X-ray frequencies.

This simple example assumes that all quasars have identical SEDs. A
more realistic calculation would take into account a spread of SEDs, which
corresponds to a range in the power-law index . The dispersion
in the power-law index
in the rest-frame ultraviolet is
substantial (Sargent, Steidel and Boksenberg 1989,
Francis et
al. 1991,
Schneider, Schmidt
and Gunn 1991), with estimates of
= 0.3-0.6.
As a consequence, our view of the luminosity function is
affected by an undesirable smoothing over absolute magnitude. Two
quasars at z = 2, with equal apparent magnitudes,
mB, and
power-law SEDs that differ in slope by = 1, have
absolute magnitudes that differ by MB = 1.2.
Equivalently, a flux-limited sample would probe 1.2 magnitudes deeper into the
luminosity function of the bluest objects relative to those with the
reddest SEDs. In a flux-limited sample, intrinsically fainter blue
quasars are preferentially brought into the sample relative to the mean SED and
associated limiting absolute magnitude. Conversely, redder
objects that are more luminous than the absolute magnitude limit are
preferentially removed. Since the number of quasars increases as the
absolute magnitude decreases, more objects are brought into the sample
than are removed. The result is to increase the number
observed (relative to a population with no dispersion in SED) and the
rate of evolution as a function of redshift is overestimated as a
consequence. Giallongo and Vagnetti (1992) made the first quantitative
investigation of the effects of a spread in SEDs on the evolution by
assuming an intrinsic spread in the power-law
slope parameter of =
0.25, and considering two models for
the evolution. Their quantitative results confirm the qualitative
expectation that the rate of evolution decreases when account
is taken of the dispersion in SEDs. Account has also been taken of the
variation in quasar SEDs by Stocke et al. (1992). In
this case a k-correction was derived specifically for BAL
quasars to place their calculated absolute magnitudes
on the same continuum-magnitude system as the control sample of
non-BALs. Figure 1 of Stocke et al. illustrates that the corrections
to the absolute magnitudes are significant, reaching M ~
0.35 at z = 2.5, in their blue passband. Most recently Francis (1993)
has considered the implications of a dispersion in SEDs on
correlations between observed properties of quasars from flux limited
samples.

If quasar SEDs can be individually characterized, then the
distribution of SEDs may be incorporated into the calculation of the
evolution explicitly as described by Warren et al. (1994) (although in
that specific investigation the quality of the spectroscopic data is
poor and their SED assignments are therefore often uncertain). Lacking
information on individual quasar SEDs, the spread in SEDs must be
included in the parametric models (e.g., Giallongo and Vagnetti
1992).